In Fig. REF  we show the nucleon specific energy \(E_0(\rho )\)  (left) and pressure \(P_0(\rho )\)  (right) in SNM adopting 5 perturbative dimensions, i.e., \(\epsilon =-2,-1,0,1\)  and 2 (corresponding to \(d\)  from 1 to 5). It is seen that as the dimension decreases, the EOS of SNM becomes much more bounded and as \(d\)  increases, the \(E_0(\rho )\)  becomes less negative.
In particular, if the \(d\)  is larger than some critical dimension, the \(E_0(\rho )\)  becomes totally positive, indicating the system is unbounded and there would be no saturation density in these dimensions. Noticing that our results given here are from perturbating the EOS in 3D, if one adopts the exact scheme, the qualitative conclusion will not change, i.e., the nucleon specific energy in SNM eventually increases as the \(d\)  increases and there would be no saturation density above certain critical dimension.
This phenomenon actually can be traced back to a deep principle behind the high-dimensional geometry: According to the relation between the Fermi momentum \(k_{\mathrm {F}}\)  and the density \(\rho \) , i.e., (REF ), one can equivalently write the density in terms of \(k_{\mathrm {F}}\)  as \(\rho =k_{\mathrm {F}}^d/[{2^{d-2}\pi ^{d/2}\Gamma (d/2+1)}]\) . The volume of a high-dimensional sphere with radius \(k_{\mathrm {F}}\)  is \((2\pi )^d\rho /4\) . For a fixed Fermi momentum \(k_{\mathrm {F}}\) , the volume of the sphere approaches zero as \(d\rightarrow \infty \)  [1]}. Inversely, if one fixes the density \(\rho \) , then the Fermi momentum \(k_{\mathrm {F}}\)  should approach infinity as \(d\rightarrow \infty \) , indicating that the kinetic EOS of SNM namely \(dE_{\mathrm {F}}/(d+2)\sim E_{\mathrm {F}}=k_{\mathrm {F}}^2/2M\)  (as \(d\rightarrow \infty \) , see (REF )) will always denominate over the potential part, irrespective of the form of the latter.
<FIGURE>